Where in Peter's sermon to Cornelius do we read anything about the Lord Jesus becoming sin for us or dying for our sins?
So you still don't believe.
Here are scriptures about Jesus dying for our sins, and note that these scriptures are from Peter, John, and Paul, and from whomever you think wrote Hebrews:
He himself bore our sins in his body on the tree, so that we might die to sins and live for righteousness; by his wounds you have been healed. 1 Peter 2:24.
Through the sacrifice of Jesus, he gives to all those that come to him freedom from the guilt, power, and punishment of sin. See Hebrews 10:4; 1 Corinthians 15:56; See 2 Peter 2:9.
Through Jesus, we have grace from God and love. Jesus puts us into perfect holiness, joy, and fellowship with himself and the Father. See Hebrews 10:10; Hebrews 1:9; 2 Corinthians 13:14.
Through Jesus’ death we are reconciled to God. See Romans 5:10.
Jesus appeared so that he might take away our sins. See 1 John 3:5.
God sent his own Son in the likeness of sinful man to be a sin offering. See Romans 8:3.
An offering for the sins of His people. See Hebrews 10:5.
The LORD has made his life a sacrifice for our wrongdoings. See Isaiah 53:10.
It was the LORD’s will to crush him with suffering. See Isaiah 53:10.
Christ, who knew no sin of his own became sin for believers, so they, who have no righteousness of their own, are made the righteousness of God through Jesus. See 2 Corinthians 5:21.
“And their sins and iniquities will I remember no more. See Hebrews 10:17.
God gives us this through Jesus Christ, “where remission of these is, there is no more offering for sin.” See Hebrews 10:18.
Jesus bore the sin of many, and made intercession for the transgressors. See Isaiah 53:12
Jesus Christ once for all has made us holy through the sacrifice of his body. Hebrews 10:10.
For God was pleased to have all his fullness dwell in him, and through him to reconcile to himself all things, whether things on earth or things in heaven, by making peace through his blood, shed on the cross. See Colossians 1:19-21.
God sanctifies us by the truth; His word is truth. John 17:17.
Jesus sanctified himself, that he may truly sanctify us. John 17:19.
Jesus was made perfect through suffering. Hebrews 2:10.
Jesus is able to sympathize with our weaknesses. Hebrews 4:15.
We have cleansing by the washing with water through the word. Ephesians 2:26.
We are also to purify ourselves from everything that contaminates body and spirit, perfecting holiness out of reverence for God. 2 Corinthians 7:1.
Jesus conquered death when God raised him from the dead. 2 Timothy 1:10.
God has given us eternal life and this life is in His Son. John 3:16.
He who has the Son has life; He who does not have the Son of God does not have life. 1 John 5:12.
All who reject Jesus will one day face God’s wrath. John 3:36
His words are life. John 6:63
Do you deny that those who believed the "good news" that Jesus is the Christ, the Son of God, were saved the moment when they believed that gospel?
No one is saved by believing alone. They must have in their heart agreed to not sin anymore. When people heard the good news, they REPENTED and were saved.
If all the jailer had to do is believe and nothing else, why did he ask what he had to do and why then was he water baptized with the baptism of repentance?
In Acts 2:37 those to whom Peter preached responded by asking, “Men and brethren, what shall we do?”
Why did Peter say, “Repent and be baptized, every one of you, in the name of Jesus Christ for the forgiveness of your sins, and you will receive the gift of the Holy Spirit.
That is two different times now you see that people asked what should they do and they repented as was evident by their water baptism.
In addition, we see that after Paul believed he was baptized with the baptism of repentance (Acts 9:18); as were the Samaritans (Acts 8:12); the Ethiopian (Acts 8:35-39); Cornelius (Acts 10:47-48) and Crispus (Acts 18:8).
Paul says that he planted the seed and that Apollos watered it.
That is about Paul speaking the message and Apollos doing the baptizing.
1 Corinthians 3:6 I planted the seed, Apollos watered it, but God has been making it grow
Just because Paul did not do a lot of water baptisms does not mean he did not teach water baptism.
1 Corinthians 10:1 For I do not want you to be ignorant of the fact, brothers and sisters, that our ancestors were all under the cloud and that they all passed through the sea. 2 They were all baptized into Moses in the cloud and in the sea.
Did you read that? Paul was speaking about water baptism.
Read what Paul says in the next two scriptures about water baptism:
Romans 6:3 Or don't you know that all of us who were baptized into Christ Jesus were baptized into his death?
Romans 6:4 We were therefore buried with him through baptism into death in order that, just as Christ was raised from the dead through the glory of the Father, we too may live a new life.
Paul is describing water baptism. The believer comes to make the pledge to God, to die to the sins of the world; so now standing in the water the believer falls back, as if dead; then, the believer goes under the water, buried; then, the believer rises up out of the water, raises up to live a new life.
We were "buried" with him through BAPTISM into death and RAISED from the dead...raised to live through Christ.